The Disjunctive Syllogism regarded as an Immediate Inference.

§ 770. If no stress be laid on the transition from disjunctive hypothesis to fact, the disjunctive syllogism will run with the same facility as its predecessor into the moulds of immediate inference.

§ 771.

Denial of Antecedent. Subalternation.

Either A is B or C is D, Every case of A not being B
is a case of C being D.
.'. A not being B, C is D. .'. Some case of A not being B
is a case of C being D.

§ 772.

Denial of Consequent. Conversion by Contraposition
+ Subalternation.

Either A is B or C is D. All cases of A not being B
are cases of C being D.
.'. C not being D, A is B .'. All cases of C not being D are
cases of A being B.
.'. Some case of C not being D is
a case of A being B.

§ 773. Similarly the two invalid types of disjunctive syllogism will be found to coincide with fallacies of immediate inference.

§ 774.

Affirmation of Antecedent. Contraposition without
Conversion.

Either A is B or C is D. All cases of A not being B are
cases of C being D.
.'. A being B, C is not D .'. All cases of A being B are
cases of C not being D.

§ 775. The affirmation of the antecedent thus comes under the formula—

All not-A is B,
.'. All A is not-B,

a form of inference which cannot hold except where A and B are known to be incompatible. Who, for instance, would assent to this?—

All non-boating men play cricket.
.'. All boating men are non-cricketers.

§ 776.

Affirmation of Consequent. Simple Conversion of A.

Either A is B or C is D. All cases of A not being B are
cases of C being D.
.'.C being D, A is not B. .'. All cases of C being D are
cases of A not being B.

§ 777. We may however argue in this way—

Conversion of A per accidens.
Either A is B or C is D. All cases of A not being B
are cases of C being D.
.'. C being D, A is sometimes B. .'. Some cases of C being D are
cases of A not being B.

The men who pass this examination must have either talent or industry.
.'. Granting that they are industrious, they may be without talent.